The problem is stated as:
Alaska has over half of all earthquakes in the United States. In particular, earthquakes with magnitude>8 on the Richter scale occur in Alaska on average every 13 years. Suppose that these occur according to a Poisson process. Compute the probability that two consecutive such earthquakes are at least 5 years apart.
My attempt:
So, if we let $X(t)$ denote the amount of earthquakes with magnitude $> 8$ in time $t$, we know that $X(t)$ is a poisson distribution with intensity $\lambda = t/13$. Since the expected value $\lambda = 1$ for $t=13$.
What is sought, is $P(X(t) - X(0) = 2 | t \geq 5)$, and by the memoryless property, we can choose our time interval as $[0,t]$ without loss of generality.
Now, from this, we know that $X(t)-X(0)$ is also poisson distributed with intensity $t/13$, meaning we are essentially searching for $P(X(t) = 2 | t \geq 5) = 1 - P(X(t) = 2 | t < 5) = 1 - \sum_{t=1}^4 P(X(t)=2) = 1- \sum_{t=1}^4 e^{-t/13} \frac{(t/13)^2}{2!} $
It seems I've complicated things since my answer sheet tells me that it's $e^{-5/13}$, and I don't really see what steps I'm doing wrong. From the answer sheet, it almost sees like they substitued the $5$ inside some formula directly, but I can't see which one it is.
I'd be glad if anyone could clarify the steps in order to solve this problem. Thanks.